Tài liệu CDMA truy cập và chuyển mạch P9 pptx

9
Nonlinear Amplification of
Synchronous CDMA
9.1 Overview
As we have described in Chapter 3, the SS/CDMA uses orthogonal CDMA for both
uplink and downlink transmission. On board the satellite, each CDMA channel is
routed to a destination downlink beam by the Code Division Switch (CDS). All
channels in the same output port of the CDS are combined and then amplified
with a Traveling Wave Tube (TWT) amplifier for downlink transmission. Due to
the large number of users in the system, the amplitude of the combined signal has
a large variance, which makes its amplification difficult since it may drive the TWT
into saturation. This phenomenon also appears in terrestrial wireless systems if the
downlink transmission at the base stations is CDMA.
Nonlinear distortions in satellites may also result for other reasons, such as
drastically physical changes in the environment, for example, temperature variations
and vibration noise. The causes of these distortions are sometimes predictable, such
as the significant temperature variations in satellites; or they might be unpredictable,
such as the variable aging of local oscillators in harsh environments. These hardware
distortions, appearing in the forms of phase noise, spurious phase modulation,
frequency offset, filter amplitude and phase ripple, data asymmetry and modulator
gain imbalance in the transmitter, as well as nonlinear amplitude and phase distortions
in the power amplifier, typically reduce the system performance from a few tenths of
dB to as large as ten dB. Among all of them, the nonlinear distortions existing in
the power amplifiers contribute to the degradation of the system. While analyses of
CDMA systems are based on the assumptions that signal waveforms are ideally linearly
retransmitted over the power amplifiers, it has been known that the existence of these
nonlinearities impacts upon the real system design.
The effects of nonlinear distortions on CDMA systems can be categorized into two
classes – out-band degradation and in-band degradation. Due to high demand on the
frequency bandwidth, stringent regulatory emission requirements have always been
enforced to prevent interference with other communication systems. To accommodate

more users simultaneously in the designated frequency bandwidth, signals transmitted
over wireless channels are always shaped so as to have a compact spectrum within
this frequency bandwidth. It also means the out-band emission has to be below
the regulated level. However, nonlinear distortions reshape the signals so that they
CDMA: Access and Switching: For Terrestrial and Satellite Networks
Diakoumis Gerakoulis, Evaggelos Geraniotis
Copyright © 2001 John Wiley & Sons Ltd
ISBNs: 0-471-49184-5 (Hardback); 0-470-84169-9 (Electronic)
212 CDMA: ACCESS AND SWITCHING
lose their compactness in spectrum which leads to out-band spectral regrowth (see
references [1] and [2]). A band-pass filter then has to be utilized before the signal
transmission in order to reject this undesired out-band power. The inefficiency of
power utilization thus results, and the filtered signal experiences higher intersymbol
interference. The BER then increases due to this extra undesired interference. We call
this out-band degradation, since the degradation is caused by the rejection of out-band
power.
The second effect of nonlinear distortions is the in-band degradation (see references
[3] and [4]). Suppose the power gained from the nonlinear amplifier is totally consumed
while signals travel through the channel. This nonlinear amplifier can then be regarded
as a nonlinear transformation between the transmitters and receivers. Note that in a
fully orthogonal system (i.e. all users are fully synchronized with a set of orthogonal
spreading codes), orthogonality is preserved only if the channel is linear. In other
words, in a non-fading linear channel, the only interference to the receiver in this
system is the thermal noise. No Multiple-Access Interference (MAI) exists. Therefore,
with or without the out-band filter, the nonlinear transformation has destroyed part
of the orthogonality of the system, which introduces MAI to an originally orthogonal
system or more MAI to an originally non-orthogonal system. The result is a higher
BER, which further leads to lower system capacity.
Aein and Pickholtz [5] presented a simple phaser model to analyze the Bit Error
Rate (BER) performance of an asynchronous CDMA system accessing a RF limiter

possessing amplitude-phase conversion (AM/PM) intermodulation effect. Their model
considered interference in the form of multiple access noise, a Continuous Wave (CW)
tone, or a combination of both. In the above reference, however, the amplitude-
to-amplitude conversion (AM/AM) intermodulation effect in the channel was not
included. Baer [6] analyzed a two-user PN spread-spectrum system with a hard limiter
in the channel. However, the effect of MAI was not considered. Note that both of
these papers focused on systems with very few users (one or two), and thus with
the major source of interference on the desired user being the self-interference due to
nonlinear distortion. Nonlinear amplifiers usually exhibit both AM/PM and AM/AM
distortions, in which the hard limiter is not the most accurate model for the AM/AM
conversion. Recently, Chen [7] analyzed the effects of nonlinearities on asynchronous
systems with MAI, but his results were only limited to the evaluation of the Signal-
to-Noise Ratio (SNR).
None of the previous work analyzed the CDMA system performance in terms of
BER and system capacity for different data modulation schemes and various spreading
codes. Neither did they provide detailed descriptions of the effects of nonlinearities
on CDMA systems. The effects of AM/AM and AM/PM distortion on the sum of a
number of DS/CDMA with M -ary PSK modulation signals have not been modeled and
analyzed in detail before, although simulations have been performed for DS/CDMA
signals with BPSK or QPSK modulation. Thus, although it is known that transponder
nonlinearities have a significant effect on the performance of DS/CDMA systems, the
performance of such systems has never been quantified in a manner that allows us to
understand how these AM/AM and AM/PM distortions affect other-user interference,
and thus how to mitigate it.
In this chapter, we evaluate the performance of synchronous M -PSK CDMA
systems in the presence of nonlinear distortions. Emphasis is placed on the modeling
NONLINEAR AMPLIFICATION 213
and analytical evaluation of the combined effects of nonlinear distortion and other-
user interference on the systems of interest. Besides the complete parametric
performance evaluation, our work also prepares the ground for developing novel

techniques for Output Back-Off (OBO) mitigation. Mitigation techniques can be
used against both the nonlinear distortion and other-user interference generated from
nonlinearities.
The application motivating this study is the satellite switched CDMA system
presented in Chapter 3. The link access of the SS/CDMA is based on the SE-
CDMA, which is a synchronous CDMA as described in Chapters 3 and 6. In this
application nonlinear distortion comes from the on-board power amplifier which is
a Traveling Wave Tube (TWT). In addition to the satellite applications, the results
presented in this chapter are also applicable to base stations in wireless terrestrial
networks.
The chapter is organized as follows. After this overview, Section 9.2 describes
the model of a synchronous M-PSK CDMA system. The mathematical model for
various nonlinear distortions, as well as their effects, are discussed afterwards. System
performance evaluated by the Gaussian approximation is presented in Section 9.3.
Section 9.4 pertains to the numerical results and simulations.
9.2 System Model
As depicted in Figure 9.1, the input signal S(t) consists of a sum of synchronous CDMA
signals from the same satellite beam (or from the same cell-sector). Each beam has
K users. The nonlinear amplifier can be seen as a nonlinear pipe, which is especially
well known in satellite communications. We further assume the downlink channel is
an Additive White Gaussian Noise (AWGN) channel with attenuation equal to the
power gained through the nonlinear amplifier. This can be easily achieved since, in the
case of unequal attenuation, its effect can be easily incorporated into the parameters
of the model of the nonlinear amplifier. This system model can be viewed as a satellite
system with high uplink power, which suggests the omission of uplink noise. In this
case, the phases of all the local oscillators might be the same, which is just a special
case of this model.
9.2.1 Transmitter
Suppose each user sends an M-PSK signal with inphase (I) and quadrature (Q)
components spread by the respective user code sequence (see Figure 9.2-A). For the k

th
user, the M-PSK data signal of I and Q components can be represented respectively as
b
(k)
I
(t)=
∞

i=−∞
b
(k)
I
[i]I
T
s
(t −iT
s
)
b
(k)
Q
(t)=
∞

i=−∞
b
(k)
Q
[i]I
T

s
(t −iT
s
)

b
(k)
I
[i],b
(k)
Q
[i]

∈


2p cos
(2m −1)π
M
,

2p sin
(2m −1)π
M

,m=1, , M

214 CDMA: ACCESS AND SWITCHING
Σ
Tx

1
Tx
2
Tx
k
Rx
1
AM/PM
AM/AM
S(t)
X(t)
Y(t)
Nonlinear Amplifier
n(t)
Figure 9.1 The system model.
with equal probability
1
M
. p is the transmitted power, T
s
is the symbol period, and
I
T
is defined as
I
T
(t)=

1if0≤ t ≤ T
0 otherwise.

The spreading code signals of the k
th
user can be represented as
c
(k)
(t)=
∞

i=−∞
c
(k)
[i]I
T
c
(t −iT
c
)
where T
c
is the chip duration and c
(k)
[i] ∈{−1, +1}. Taking into account the phase
of each user’s local oscillator, the output signal of user k can be represented as
S
(k)
(t)=b
(k)
I
(t)c
(k)

(t)cos

ω
c
t + θ
(k)

+ b
(k)
Q
(t)c
(k)
(t)sin

ω
c
t + θ
(k)

= A
(k)
(t)cos(ω
c
t)+B
(k)
(t) sin(ω
c
t)
where
A

(k)
(t)=b
(k)
I
(t)c
(k)
(t)cosθ
(k)
+b
(k)
Q
(t)c
(k)
(t)sinθ
(k)
B
(k)
(t)=−b
(k)
I
(t)c
(k)
(t)sinθ
(k)
+b
(k)
Q
(t)c
(k)
(t)cosθ

(k)
Since this is a synchronous system, the transmitted signal is the sum of the individual
signal with perfect time alignment. Therefore,
S(t)=
K

k=1
S
(k)
(t)=V (t)cos(ω
c
t −Φ(t))
NONLINEAR AMPLIFICATION 215
~
π/2
Σ
Spreading
Code
Generator
)t(
(k)
I
b
)t(
(k)
Q
b
)+tcos(
)k(
c

θω
)+tsin(
)k(
c
θω
)t(S
)k(
A.
B.
T
s
∫
0
s
T
1
s
T=t
T
s
∫
0
s
T
1
s
T=t
Decision
Device
)tcos()t(c2

c
)1(
ω
)tsin()t(c2
c
)1(
ω
)t(n+)t(y
Figure 9.2 The transmitter (A), and receiver (B) models.
where
V (t)=





K

k=1
A
(k)
(t)

2
+

K

k=1
B

(k)
(t)

2
Φ(t)=tan
−1


K
k=1
B
(k)
(t)

K
k=1
A
(k)
(t)

9.2.2 Nonlinear Amplifiers
The most commonly used AM/AM model for nonlinear amplifiers is the Q function
(see Figure 9.3-A). Almost all the smooth limiters of interest preserve this shape.
The drive power at which the output power saturates is called the input saturation
power. In Figure 9.3-A, the corresponding baseband input amplitude is ±2.3. The
ratio of input saturation power to desired drive power is called the input back-
off (IBO). Similarly, the output saturation power is the maximum output power
of an amplifier. Its corresponding baseband output amplitude in Figure 9.3-A is
±1. Output back-off (OBO) is therefore the ratio of the output saturation power
to the actual output power. Increasing IBO or OBO leads to less output power,

but reduces the nonlinearities introduced during signal amplification. The trade-off
between lower power and more nonlinearities results in the highest effective SNR or,
equivalently, the highest effective E
s
/N
o
and the lowest BER. Note that increasing
OBO reduces the efficiency of amplifier power usage, which is particularly undesirable
in satellite communications. OBO can thus be regarded as the immunity strength
of a communication system to nonlinearity. The higher OBO that is required, the
216 CDMA: ACCESS AND SWITCHING
more vulnerable the system is to nonlinear distortion and the less efficient in power
usage.
However, since the Q function is not an analytical function, in order to evaluate the
system performance, we may only use computer simulation. The hard limiter model,
on the other hand, although simple is not an accurate representation of our system.
Furthermore, the hard limiter model will not be able to determine the IBO or OBO
of a nonlinear amplifier. In addition to the above two extreme options, i.e. Q function
and hard limiter, Chen [7], Forsey [8] and Kunz [9] have proposed different models. Our
model is based on both Chen [7] and Kunz [9] for reasons of accuracy and simplicity
in the evaluation of the Signal-to-Noise Ratio (SNR) of the CDMA system.
In the particular model we use, AM/AM is represented by a third-order polynomial,
which is a memoryless nonlinearity affecting only the amplitude of the input signal.
The coefficients of the polynomial are determined by placing the local maximum or
minimum at the saturation point of the nonlinear amplifier. In the meantime, AM/PM
introduces a phase distortion which is proportional to the square of the envelope of
the input signal. In other words, AM/PM: Θ[V (t)] = ηV
2
(t), η  1, and AM/AM:
V


(t)=α
1
V (t)+α
3
V
3
(t), where η, α
1
and α
3
are the corresponding parameters.
Then the input signal to the receiver is
z(t)=V

(t)cos{ω
c
t −Φ(t)+Θ[V (t)]} + n(t)
=[S
I
(t)+D
I
(t)+n
I
(t)] cos(ω
c
t)+[S
Q
(t) −D
Q

(t)+n
Q
(t)] sin(ω
c
t)
where
n(t)=n
I
(t)cos(ω
c
t)+n
Q
(t) sin(ω
c
t)
S
I
(t)=

α
1
+ α
3
V
2
(t)

K

k=1

A
(k)
(t)
D
I
(t)=η

α
1
V
2
(t)+α
3
V
4
(t)

K

k=1
B
(k)
(t)
S
Q
(t)=

α
1
+ α

3
V
2
(t)

K

k=1
B
(k)
(t)
D
Q
(t)=η

α
1
V
2
(t)+α
3
V
4
(t)

K

k=1
A
(k)

(t)
9.3 Performance Analysis
The effects of nonlinearities on CDMA signals are evaluated by analyzing their
distortions on the first two moments of the received signal. Since there is a large
number of users in the system (32 to 64), it is important that the methodology
we use is very accurate. The proposed methodology is based on so-called Gaussian
Approximation (GA). Note that to achieve accurate results, we apply the GA after
the nonlinear effect. In other words, we analyze the first two moments of the signal
after the nonlinear distortion instead of analyzing them before the distortion and then
nonlinearly transforming them to analyze the performance.
NONLINEAR AMPLIFICATION 217
Figure 9.3 (A) The amplitude transfer (AM/AM) curve and (B) the phase transfer
(AM/PM) curve.
218 CDMA: ACCESS AND SWITCHING
The distortion on the first moment is sometimes called constellation warping.The
received constellation points are no longer at their original grids due to the distortion
of amplitude and phase. Amplitude distortion changes the distance from the original
to the constellation points, while phase distortion changes their angles. Constellation
warping leads to the undesired preferences of some of the constellation points, which
means that the detection is no longer maximum likelihood. The amplitude shrinkage
also reduces the effective E
b
/N
o
.
The distortion on the second moment is called cloud forming.AsinSS/CDMA,in
most synchronous CDMA systems, Multiple-Access Interference (MAI) is minimized
by utilizing orthogonal spreading codes. We assume that users within each beam are
distinguished by orthogonal codes, while between beams we use PN-codes. Therefore,
since all downlink transmissions are perfectly synchronized, the same-beam MAI shall

be zero and the other-beam MAI is a function of the cross-correlation functions of PN-
codes. This suggests that the BER performance is a function of the received user power,
the thermal noise figure and the other-beam MAI. Note that both the orthogonality
and cross-correlation functions are the second moment of code functions. However,
in the presence of nonlinear distortions of amplitude and phase, the second-moment
distortion generates high-order moments of code functions. These high-order moments
destroy the orthogonality between same-beam spreading codes, which leads to nonzero
same-beam MAI. The other-beam MAI now also consists of second moments and
higher moments of code functions. The nonlinear MAI, composed of both same-beam
and other-beam MAIs, becomes a function of the second or even third power of the
received user power, the thermal noise figure, the high-order moments of code cross-
correlation functions and, worst of all, of the number of active users in the system.
System capacity thus reduces due to nonlinearities.
In general, amplitude nonlinearity leads to the generation of harmonics and
amplitude cross-modulation. Nonlinear phase characteristics lead to phase cross-
modulation. The effects of these two nonlinearities on the sum of CDMA signals
will force warping of signal constellations as well as the intra- and inter-beam cross-
correlation of signals.
The receiver model is shown in Figure 9.2-B. Without loss of generality, we consider
the performance of the first user. Then the output of the in-phase receiver can be
represented as
Z
I
=
1
T
s

T
s

0
y(t)c
(1)
I
(t)2 cos(ω
c
t)dt
The integrator will then reject the high frequency portion of the signal. Hence,
Z
I
= S
I
+ D
I
+ N
I
where
S
I
=
1
T
s

T
s
0
S
I
(t)c

(1)
I
(t)dt
D
I
=
1
T
s

T
s
0
D
I
(t)c
(1)
I
(t)dt
The mean of N
I
is 0 and the variance of N
I
is σ
2
N
=
N
0
T

s
.
N
0
2
is the two-sided power
spectral density.
NONLINEAR AMPLIFICATION 219
Suppose b
(1)
I
[0] = d
I
and b
(1)
Q
[0] = d
Q
. We also assume synchronization is perfect,
which means θ
(1)
=0.ThusA
(1)
[0] = d
I
c
(1)
[n], B
(1)
[0] = d

Q
c
(1)
[n], n =0 N − 1,
where A
(1)
[0] represents the value of A
(1)
(t) during the time interval from t =0
to t = T
s
. The same rule, i.e. ‘[0]’ representing t =0 T
s
, also applies to all other
functions. Therefore,
S
I
=
1
N
N−1

n=0
S
I
[0]c
(1)
I
[n]
=

1
N
N−1

n=0


α
1
+ α
3
V
2
[0]

K

k=1
A
(k)
[0]

c
(1)
I
[n]
= α
1
1
N

N−1

n=0
K

k=1
A
(k)
[0]c
(1)
I
[n]+α
3
1
N
N−1

n=0
K

k
1
,k
2
,k
3
=1
A
(k
1

)
[0]A
(k
2
)
[0]A
(k
3
)
[0]c
(1)
I
[n]
+ α
3
1
N
N−1

n=0
K

k
1
,k
2
,k
3
=1
A

(k
1
)
[0]B
(k
2
)
[0]B
(k
3
)
[0]c
(1)
I
[n]
Similarly,
D
I
=
1
N
N−1

n=0
D
I
[0]c
(1)
I
[n]

=
1
N
N−1

n=0

η

α
1
V
2
[0]+α
3
V
4
[0]

K

k=1
B
(k)
[0]

c
(1)
I
[n]

= ηα
1
1
N
N−1

n=0
K

k
1
,k
2
,k
3
=1
B
(k
1
)
[0]A
(k
2
)
[0]A
(k
3
)
[0]c
(1)

I
[n]
+ ηα
1
1
N
N−1

n=0
K

k
1
,k
2
,k
3
=1
B
(k
1
)
[0]B
(k
2
)
[0]B
(k
3
)

[0]c
(1)
I
[n]
+ ηα
3
1
N
N−1

n=0
K

k
1
,k
2
k
3
,k
4
,k
5
=1

B
(k
1
)
[0]A

(k
2
)
[0]A
(k
3
)
[0]A
(k
4
)
[0]A
(k
5
)
[0]
+2B
(k
1
)
[0]B
(k
2
)
[0]B
(k
3
)
[0]A
(k

4
)
[0]A
(k
5
)
[0]
+B
(k
1
)
[0]B
(k
2
)
[0]B
(k
3
)
[0]B
(k
4
)
[0]B
(k
5
)
[0]

c

(1)
I
[n]
The detailed derivations are shown in Appendix 9A. For the case of QPSK, the
following results can be obtained for the I component. General results for M-PSK
cases are shown in Appendix 9A. The corresponding expression for the Q component
is obtained by interchanging I to Q and Q to I in all terms:
220 CDMA: ACCESS AND SWITCHING
Y
I
= α
1
d
I
+ α
3
d
3
I
+ α
3
d
I
d
2
Q
+ ηα
1
d
3

Q
+ ηα
1
d
2
I
d
Q
+ ηα
3
d
5
Q
+2ηα
3
d
2
I
d
3
Q
+ ηα
3
d
4
I
d
Q
U
I

= a
2
K
2
+ a
1
K + a
0
σ
2
I
= b
3
K
3
+ b
2
K
2
+ b
1
K + b
0
+ σ
2
N
σ
2
IQ
=4α

1
α
3
d
I
d
Q
(K −1)R
2
p
where
a
2
=24α
3
ηd
Q
p
2
a
1
= −60α
3
ηd
Q
p
2
+

4α

3
d
I
+4α
1
ηd
Q
+12α
3
ηd
3
Q
+12α
3
ηd
2
I
d
Q

p
a
0
=36α
3
ηd
Q
p
2
−


12α
3
ηd
2
I
d
Q
+4α
3
d
I
+4α
1
ηd
Q
+12α
3
ηd
3
Q

p
b
3
=

16α
2
3

R
2
+
19
3
α
2
3
ˆ
R
2

p
3
b
2
=

−60α
2
3
R
2
− 38α
2
3
ˆ
R
2


p
3
+

12α
2
3
d
2
Q
R
2
+8α
1
α
3
R
2
+44α
2
3
d
2
I
R
2

p
2
b

1
=

72α
2
3
R
2
+
209
3
α
2
3
ˆ
R
2

p
3
+

2α
2
3
d
2
I
− 120α
2

3
d
2
I
R
2
− 20α
1
α
3
R
2
− 32α
2
3
d
2
Q
R
2
+2α
2
3
d
2
Q

p
2
+


α
2
3
R
2
d
4
Q
+ α
2
1
R
2
+6α
1
α
3
R
2
d
2
I
+2α
1
α
3
R
2
d

2
Q
+10α
2
3
d
2
I
d
2
Q
R
2
+9α
2
3
R
2
d
4
I

p
b
0
=

−28α
2
3

R
2
− 38α
2
3
ˆ
R
2

p
3
+

12α
1
α
3
R
2
+76α
2
3
d
2
I
R
2
+20α
2
3

d
2
Q
R
2
− 2α
2
3
d
2
Q
− 2α
2
3
d
2
I

p
2
+

−α
2
1
R
2
− 6α
1
α

3
R
2
d
2
I
− 2α
1
α
3
R
2
d
2
Q
− 9α
2
3
R
2
d
4
I
− 10α
2
3
d
2
I
d

2
Q
R
2
−α
2
3
R
2
d
4
Q

p
σ
2
I
is the variance of the I component interference, and σ
2
IQ
is the covariance of the
I and Q components. S
I
= S
I
and D
I
= D
I
. Y

I
+ U
I
= S
I
+ D
I
. Y
I
is the part of
S
I
+ D
I
from only the first user. All other terms, regarded as a disturbance of the first
moment from other users, are represented by U
I
. Note that a non-zero-mean cross-
correlated Gaussian interference results due to nonlinearities. All the corresponding
results for the Q components can be easily obtained by exchanging I and Q and Q
and I in the expressions of I the component.
To compute the probability of symbol error, let φ
m
=tan
−1
Y
I
+U
I
Y

Q
+U
Q
and m
correspond to different d
I
and d
Q
,where
(d
I
,d
Q
)∈


2p cos
(2m −1)π
M
,

2p sin
(2m −1)π
M

,m=1, , M

Furthermore, let
ρ
I,Q

=
E{(Z
I
− Z
I
)(Z
Q
− Z
Q
)}
σ
I
σ
Q
NONLINEAR AMPLIFICATION 221
For equi-probable M-ary PSK symbols, the probability of error is (see Appendix 9A
for details)
P
e
=
1
M
M

m=1
P
e|m
where
P
e|m

=1−

φ
m
+
π
M
φ
m
−
π
M
¯
f(φ, φ
m
)dφ
¯
f(φ, φ
m
) is defined as
¯
f(φ, φ
m
)=
1

2πσ
I
σ
Q


1 −ρ
I,Q
sin(2φ)
cos(φ −φ
m
) −ρ
I,Q
sin(φ + φ
m
)
1 −ρ
I,Q
sin(2φ)
· exp



−
[1 −ρ
I,Q
sin(2φ
m
)] −
[cos(φ−φ
m
)−ρ
I,Q
sin(φ+φ
m

)]
2
1−ρ
I,Q
sin(2φ)
2(1 −ρ
2
I,Q
)σ
I
σ
Q



The effects on the spread-spectrum signals after despreading can be summarized as
1. Cloud forming: each constellation point becomes a cloud due to linear and
nonlinear ISI, as well as intra- and inter-user cross-modulation of the I and
Q components, and
2. Signal warping: the respective centroid of the clouds (i.e. the received
constellation points) are no longer at the original position of the constellation
points.
From another point of view, nonlinearities shrink the constellation and transform its
power into clouds. So besides the filtered out-of-band power, nonlinearities further
reduce the effective signal power.
9.4 Numerical Results
We first consider a system with 64 chips per code. Three different code families
(i.e. OG, PG and PN) are investigated along with BPSK, QPSK and 8-PSK data
modulation schemes. The AM/AM curve of the nonlinear amplifier is shown as a
dotted line in Figure 9.3-A, while the AM/PM curve is shown in Figure 9.3-B. The

way to compute the input power, output power and IBO of a nonlinear amplifier is
shown in Appendix 9A.5.
Figure 9.4 shows the IBO versus E
s
/N
0
and effective E
s
/N
0
. It shows that the
E
s
/N
0
(dashed line) increases as IBO decreases, because less IBO means more power
is being transmitted. On the other hand, as IBO decreases the effective E
s
/N
0
(solid
line) deviates from the E
s
/N
0
(dashed line) to a lower position beginning from about
8 dB. It even descends after the IBO is less than 3 dB. This behavior reveals that
the effects of nonlinearities start emerging at 8 dB. They further deteriorate, and thus
dominate the output signals after IBO is less than 3 dB.
Figure 9.5 shows the SER for 32 users, and Figure 9.6 shows that for 64 users.

As shown, the effective E
s
/N
0
begins lowering, and after 3 dB IBO decreases. This
concludes that in this particular system considered, the optimum operating point of
the nonlinear amplifier is around 3 dB IBO.
222 CDMA: ACCESS AND SWITCHING
Figure 9.4 The E
s
/N
0
versus the Input-Back-Off (IBO) in db.
Figure 9.5 The Symbol Error Rate (SER) versus the Input Back-Off (IBO) in db
with 32 users and for orthogonal (OG), preferred-phased (PG) and
pseudonoise (PN) sequences.
NONLINEAR AMPLIFICATION 223
Figure 9.6 The Symbol Error Rate (SER) versus the Input Back-Off (IBO) in db
with 64 users and for orthogonal (OG), perferred-phased (PG) and
pseudonoise (PN) sequences.
Figure 9.7 Comparisons between analysis and simulation of the SER vs. the IBO for
orthogonal (OG) sequences and for QPSK and 8-PSK.
224 CDMA: ACCESS AND SWITCHING
To compare with these analytical results, computer simulations were conducted for
systems with QPSK and 8-PSK data modulation schemes and OG spreading code.
AM/AM nonlinear distortion is represented as the solid line in Figure 9.3-A. The
results are shown in Figure 9.7. Their solid consistency further assures the validity of
the analytical results.
According to the analytical and numerical results obtained, several conclusions are
made which are summarized as follows:

1. The received constellation points are no longer at their original places.
2. Given an amplifier, for a fixed E
b
/N
0
requirement, different number of active
users in a system has different Output Back-Off (OBO) values.
3. The value of the OBO as well as the interference and received constellation
points are implicit functions of the number of active users.
4. Nonlinearities destroy the orthogonality of the inter-user same-phase
components.
5. Nonlinearities leads to a non-zero-mean and cross-correlated jointly Gaussian
interference in the I and Q branches of the receiver.
9.5 Conclusions
The performances of synchronous M-PSK CDMA systems with nonlinear distortions
were evaluated analytically. The effects of nonlinearities appear more significantly
while the back-off is less. The trade-off between power and nonlinearities, which
achieves maximal effective E
s
/N
0
, determines the desired operating point of a
nonlinear amplifier. The performance of BPSK data modulation scheme, as expected,
is better than QPSK, which is even better than 8-PSK. The OG and PG spreading
codes have almost the same performance and both of them outperform the PN
spreading code.
References
[1] S W. Chen, W. Panton and R. Gilmore ‘Effects of Nonlinear Distortion on
CDMA Communication Systems’ IEEE MTT-S Digest, 1996, pp. 775–778.
[2] N. I. Smirnov and S. F. Gorgadze ‘An Estimate of the Power Utilization

Efficiency of a Nonlinear Tranponder in Data Transmission Systems with
Code Division Multiplexing’ Elektrosvyaz, June 1995, pp. 21–24.
[3] Pen Li and E. Geraniotis ‘Effects of Nonlinear Distortion on Synchronous M-
PSK DS/CDMA Systems’ Conference on Information Science and Systems,
Baltimore, MD, March 1997, pp. 966–971.
[4] Pen Li and E. Geraniotis ‘Performance Analysis of Synchronous M-PSK
DS/CDMA Multi-Tier System with a Nonlinear Amplifier’ IEEE Symposium
on Computers and Communications, Alexandria, Egypt, July 1997, pp. 275–
279.
[5] J. M. Aein and R. L. Pickholtz ‘A Simple Unified Phasor Analysis for
PN Multiple Access to Limiting Repeaters’ IEEE Trans. on Commun.,
Vol. COM-30, No. 5, May 1982, pp. 1018–1026.
NONLINEAR AMPLIFICATION 225
[6] H. P. Baer ‘Interference Effects of Hard Limiting in PN Spread-Spectrum
Systems’ IEEE Trans. on Commun., Vol. COM-30, No. 5, May 1982,
pp. 1010–1017.
[7] L. Chen, S. Ikuta, M. Kominami and H. Kusaka ‘An Analysis of Nonlinear
Distortion due to TWT in Asynchronous DS-SSMA Communication
Systems’ IEEE Spread Spectrum Commun., 1994, pp. 279–302.
[8] R. J. Forsey, V. E. Gooding, P. J. McLane and L. L. Campbell ‘M-ary PSK
Transmission via a Coherent Two-Link Channel Exhibiting AM-AM and
AM-PM Nonlinearities’ IEEE Trans. on Commun., Vol. COM-26, No. 1,
January 1978, pp. 116–123.
[9] W.E. Kunz, J. H. Foster and R. F. Lazzarini ‘Travelling-Wave Tube
Amplifier Characteristics for Communications’ The Microwave Journal,
March 1967, pp. 41–46.
Appendix 9A: Performance Evaluation
9A.1 Evaluation of the First Moment
Let
E

θ
[·]=
1
(2π)
K−1

2π
0

2π
0
···

2π
0
·dθ
K
···dθ
3
dθ
2
By knowing the fact that
E
θ

A
(k)
[0]

=


A
(1)
[0] if k =1;
0 otherwise
E
θ

A
(k
1
)
[0]A
(k
2
)
[0]

=





A
(1)
[0]
2
k
1

= k
2
=1;
1
2

b
(k)
I
[0]
2
+ b
(k)
Q
[0]
2

k
1
= k
2
= k =1;
0 otherwise
E
θ

A
(k
1
)

[0]A
(k
2
)
[0]A
(k
3
)
[0]

=









A
(1)
[0]
3
k
1
= k
2
= k
3

=1;
1
2
A
(1)
[0]

b
(k)
I
[0]
2
+ b
(k)
Q
[0]
2

if one of k
1
,k
2
, or k
3
=1
and the other two are equal to k;
0 otherwise
E
θ


A
(k
1
)
[0]B
(k
2
)
[0]B
(k
3
)
[0]

=





A
(1)
[0]B
(1)
[0]
2
k
1
= k
2

= k
3
=1;
1
2
A
(1)
[0]

b
(k)
I
[0]
2
+ b
(k)
Q
[0]
2

k
1
=1,k
2
= k
3
= k =1;
0 otherwise
Then
K


k
1
=1
K

k
2
=1
ηA
(k
1
)
[0]A
(k
2
)
[0] = d
2
I
+
1
2
K

k=2

b
(k)
I

[0]
2
+ b
(k)
Q
[0]
2

226 CDMA: ACCESS AND SWITCHING
K

k
1
=1
K

k
2
=1
K

k
3
=1
E
θ

A
(k
1

)
[0]A
(k
2
)
[0]A
(k
3
)
[0]

= A
(1)
[0]
3
+
3
2
K

k=2
A
(1)
[0]

b
(k)
I
[0]
2

+ b
(k)
Q
[0]
2

K

k
1
=1
K

k
2
=1
K

k
3
=1
E
θ

A
(k
1
)
[0]B
(k

2
)
[0]B
(k
3
)
[0]

= A
(1)
[0]B
(1)
[0]
2
+
1
2
K

k=2
A
(1)
[0]

b
(k)
I
[0]
2
+b

(k)
Q
[0]
2

Therefore, from the expression of S
I
(t) in Section 9.2.2.
E
θ
[S
I
]=α
1
1
N
N−1

n=0
A
(1)
[0]c
(1)
I
[n]
+ α
3
1
N
N−1


n=0

A
(1)
[0]
3
+
3
2
K

k=2
A
(1)
[0]

b
(k)
I
[0]
2
+ b
(k)
Q
[0]
2


c

(1)
I
[n]
+ α
3
1
N
N−1

n=0

A
(1)
[0]B
(1)
[0]
2
+
1
2
K

k=2
A
(1)
[0]

b
(k)
I

[0]
2
+b
(k)
Q
[0]
2


c
(1)
I
[n]
= α
1
d
I
+α
3
d
3
I
+α
3
d
I
d
2
Q
+2α

3
d
I
K

k=2

b
(k)
I
[0]
2
+ b
(k)
Q
[0]
2

The first moment of D
I
can be obtained similarly.
9A.2 Evaluation of the Second Moment
To compute the variance, we rewrite the expression of S
I
(t) in Section 9.2.2 as follows:
S
I
= α
1
K


k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)
+ α
1
K

k=1
C(I
1
,Q
k
)(k)b
(k)
Q
[0] sin θ
(k)
+ α
3
Ψ
where
C(E

i
,F
j
)=
1
N
N−1

n=0
c
(i)
E
[n]c
(j)
F
[n],E,F∈{I,Q}
NONLINEAR AMPLIFICATION 227
Ψ=
K

k
1
=1
K

k
2
=1
K


k
3
=1

a
7
(k
1
,k
2
,k
3
)cosθ
(k
1
)
cos θ
(k
2
)
cos θ
(k
3
)
+ a
6
(k
1
,k
2

,k
3
)cosθ
(k
1
)
cos θ
(k
2
)
sin θ
(k
3
)
+ a
5
(k
1
,k
2
,k
3
)cosθ
(k
1
)
sin θ
(k
2
)

cos θ
(k
3
)
+ a
4
(k
1
,k
2
,k
3
)cosθ
(k
1
)
sin θ
(k
2
)
sin θ
(k
3
)
+ a
3
(k
1
,k
2

,k
3
)sinθ
(k
1
)
cos θ
(k
2
)
cos θ
(k
3
)
+ a
2
(k
1
,k
2
,k
3
)sinθ
(k
1
)
cos θ
(k
2
)

sin θ
(k
3
)
+ a
1
(k
1
,k
2
,k
3
)sinθ
(k
1
)
sin θ
(k
2
)
cos θ
(k
3
)
+a
0
(k
1
,k
2

,k
3
)sinθ
(k
1
)
sin θ
(k
2
)
sin θ
(k
3
)

Also, a
i
(k
1
,k
2
,k
3
)’s are defined as
a
7
(k
1
,k
2

,k
3
)=C(I
k
1
,I
k
2
,I
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
I
[0]b
(k
3
)
I
[0]
+ C(I

k
1
,Q
k
2
,Q
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
Q
[0]b
(k
3
)
Q
[0]
a
6
(k
1

,k
2
,k
3
)=C(I
k
1
,I
k
2
,Q
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
I
[0]b
(k
3
)
Q

[0]
− C(I
k
1
,Q
k
2
,I
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
Q
[0]b
(k
3
)
I
[0]
a
5

(k
1
,k
2
,k
3
)=C(I
k
1
,Q
k
2
,I
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
Q
[0]b
(k
3

)
I
[0]
− C(I
k
1
,I
k
2
,Q
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
I
[0]b
(k
3
)
Q
[0]

a
4
(k
1
,k
2
,k
3
)=C(I
k
1
,Q
k
2
,Q
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
Q
[0]b

(k
3
)
Q
[0]
+ C(I
k
1
,I
k
2
,I
k
3
,I
1
)b
(k
1
)
I
[0]b
(k
2
)
I
[0]b
(k
3
)

I
[0]
a
3
(k
1
,k
2
,k
3
)=C(Q
k
1
,I
k
2
,I
k
3
,I
1
)b
(k
1
)
Q
[0]b
(k
2
)

I
[0]b
(k
3
)
I
[0]
+ C(Q
k
1
,Q
k
2
,Q
k
3
,I
1
)b
(k
1
)
Q
[0]b
(k
2
)
Q
[0]b
(k

3
)
Q
[0]
a
2
(k
1
,k
2
,k
3
)=C(Q
k
1
,I
k
2
,Q
k
3
,I
1
)b
(k
1
)
Q
[0]b
(k

2
)
I
[0]b
(k
3
)
Q
[0]
− C(Q
k
1
,Q
k
2
,I
k
3
,I
1
)b
(k
1
)
Q
[0]b
(k
2
)
Q

[0]b
(k
3
)
I
[0]
a
1
(k
1
,k
2
,k
3
)=C(Q
k
1
,Q
k
2
,I
k
3
,I
1
)b
(k
1
)
Q

[0]b
(k
2
)
Q
[0]b
(k
3
)
I
[0]
− C(Q
k
1
,I
k
2
,Q
k
3
,I
1
)b
(k
1
)
Q
[0]b
(k
2

)
I
[0]b
(k
3
)
Q
[0]
a
0
(k
1
,k
2
,k
3
)=C(Q
k
1
,Q
k
2
,Q
k
3
,I
1
)b
(k
1

)
Q
[0]b
(k
2
)
Q
[0]b
(k
3
)
Q
[0]
+ C(Q
k
1
,I
k
2
,I
k
3
,I
1
)b
(k
1
)
Q
[0]b

(k
2
)
I
[0]b
(k
3
)
I
[0]
where
C(E
l
1
,F
l
2
,G
l
3
,H
l
3
)=
1
N
N−1

n=0
c

(l
1
)
E
[n]c
(l
2
)
F
[n]c
(l
3
)
G
[n]c
(l
4
)
H
[n]
228 CDMA: ACCESS AND SWITCHING
where E, F,G,H ∈{I,Q}.NotethatS
I
and D
I
are uncorrelated and η  1. So
η(S
I
+ D
I

)
2
= ηS
I
2
+2S
I
D
I
+ D
I
2
≈α
2
1
(Υ
1
+Υ
2
)+2α
1
α
3
(Υ
3
+Υ
4
)+α
3
3

ηΨ
where
Υ
1
= η

K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)

2
Υ
2
= η

K

k=1
C(I
1
,Q

k
)b
(k)
Q
[0] sin θ
(k)

2
Υ
3
= ηΨ
K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)
Υ
4
= ηΨ
K

k=1
C(I

1
,Q
k
)b
(k)
Q
[0] sin θ
(k)
For 1 ≤ k
0
,k
1
,k
2
,k
3
≤ K
ηcos θ
(k
0
)
cos θ
(k
1
)
cos θ
(k
2
)
sin θ

(k
3
)
= ηcos θ
(k
0
)
cos θ
(k
1
)
sin θ
(k
2
)
cos θ
(k
3
)
=0
ηcos θ
(k
0
)
sin θ
(k
1
)
cos θ
(k

2
)
cos θ
(k
3
)
= ηcos θ
(k
0
)
sin θ
(k
1
)
sin θ
(k
2
)
sin θ
(k
3
)
=0
ηsin θ
(k
0
)
cos θ
(k
1

)
cos θ
(k
2
)
cos θ
(k
3
)
= ηsin θ
(k
0
)
cos θ
(k
1
)
sin θ
(k
2
)
sin θ
(k
3
)
=0
ηsin θ
(k
0
)

sin θ
(k
1
)
cos θ
(k
2
)
sin θ
(k
3
)
= ηsin θ
(k
0
)
sin θ
(k
1
)
sin θ
(k
2
)
cos θ
(k
3
)
=0
After some computations, we can get

Υ
1
= η

K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)

2
= d
2
I
+
1
2
K

k=2
C(I
1
,I

k
)
2
b
(k)
I
[0]
2
Υ
2
= η

K

k=1
C(I
1
,Q
k
)b
(k)
Q
[0] sin θ
(k)

2
=
1
2
K


k=2
C(I
1
,Q
k
)
2
b
(k)
Q
[0]
2
η

K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)

·


K

k
1
=1
K

k
2
=1
K

k
3
=1
a
7
(k
1
,k
2
,k
3
)cosθ
(k
1
)
cos θ
(k
2

)
cos θ
(k
3
)

NONLINEAR AMPLIFICATION 229
= d
4
I
+ d
2
I
d
2
Q
+
K

k=2

3
2
d
2
I
b
(k)
I
[0]

2
+
3
2
C(I
1
,I
k
)
2
d
2
I
b
(k)
I
[0]
2
+
1
2
d
2
I
b
(k)
Q
[0]
2
+ C(I

1
,I
k
,Q
1
,Q
k
)d
I
d
Q
b
(k)
I
[0]b
(k)
Q
[0] + C(I
1
,I
k
)C(Q
1
,Q
k
)d
I
d
Q
b

(k)
I
[0]b
(k)
Q
[0]
+
1
2
C(I
1
,I
k
)
2
d
2
Q
b
(k)
I
[0]
2
+
3
8
C(I
1
,I
k

)
2
b
(k)
I
[0]
4
+
3
8
C(I
1
,I
k
)
2
b
(k)
I
[0]
2
b
(k)
Q
[0]
2

+

2≤k,l≤K

k=l

3
4
C(I
1
,I
k
)
2
b
(k)
I
[0]
2
b
(l)
I
[0]
2
+
1
4
C(I
1
,I
k
)
2
b

(k)
I
[0]
2
b
(l)
Q
[0]
2
+
1
2
C(I
1
,I
k
)C(I
1
,I
l
,Q
k
,Q
l
)b
(k)
I
[0]b
(l)
I

[0]b
(k)
Q
[0]b
(l)
Q
[0]

η

K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)

·

K

k
1
=1

K

k
2
=1
K

k
3
=1
a
4
(k
1
,k
2
,k
3
)cosθ
(k
1
)
sin θ
(k
2
)
sin θ
(k
3
)


=
K

k=2

1
2
d
2
I
b
(k)
I
[0]
2
+
1
2
d
2
I
b
(k)
Q
[0]
2
+
1
8

C(I
1
,I
k
)
2
b
(k)
I
[0]
2
b
(k)
Q
[0]
2
+
1
8
C(I
1
,I
k
)
2
b
(k)
I
[0]
4


+

2≤k,l≤K
k=l

1
4
C(I
1
,I
k
)
2
b
(k)
I
[0]
2
b
(l)
I
[0]
2
+
1
4
C(I
1
,I

k
)
2
b
(k)
I
[0]
2
b
(l)
Q
[0]
2

η

K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)

·


K

k
1
=1
K

k
2
=1
K

k
3
=1
a
2
(k
1
,k
2
,k
3
)sinθ
(k
1
)
cos θ
(k

2
)
sin θ
(k
3
)

=
K

k=2

1
2
d
2
I
b
(k)
Q
[0]
2
−
1
2
C(I
1
,I
k
,Q

1
,Q
k
)d
I
d
Q
b
(k)
I
[0]b
(k)
Q
[0]

+

2≤k,l≤K
k=l

1
4
C(I
1
,I
k
)
2
b
(k)

I
[0]
2
b
(l)
Q
[0]
2
−
1
4
C(I
1
,I
k
)C(I
1
,I
l
,Q
k
,Q
l
)b
(k)
I
[0]b
(l)
I
[0]b

(k)
Q
[0]b
(l)
Q
[0]

η

K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)

·

K

k
1
=1
K


k
2
=1
K

k
3
=1
a
1
(k
1
,k
2
,k
3
)sinθ
(k
1
)
sin θ
(k
2
)
cos θ
(k
3
)


230 CDMA: ACCESS AND SWITCHING
=
K

k=2

1
2
d
2
I
b
(k)
Q
[0]
2
−
1
2
C(I
1
,I
k
,Q
1
,Q
k
)d
I
d

Q
b
(k)
I
[0]b
(k)
Q
[0]

+

2≤k,l≤K
k=l

1
4
C(I
1
,I
k
)
2
b
(k)
I
[0]
2
b
(l)
Q

[0]
2
−
1
4
C(I
1
,I
k
)C(I
1
,I
l
,Q
k
,Q
l
)b
(k)
I
[0]b
(l)
I
[0]b
(k)
Q
[0]b
(l)
Q
[0]


η

K

k=1
C(I
1
,Q
k
)b
(k)
Q
[0] sin θ
(k)

·

K

k
1
=1
K

k
2
=1
K


k
3
=1
a
6
(k
1
,k
2
,k
3
)cosθ
(k
1
)
cos θ
(k
2
)
sin θ
(k
3
)

=
K

k=2

1

2
C(I
1
,Q
k
)
2
d
2
I
b
(k)
Q
[0]
2
−
1
2
C(I
1
,Q
k
)C(Q
1
,I
k
)d
I
d
Q

b
(k)
I
[0]b
(k)
Q
[0]

+

2≤k,l≤K
k=l

1
4
C(I
1
,Q
k
)
2
b
(l)
I
[0]
2
b
(k)
Q
[0]

2
−
1
4
C(I
1
,Q
k
)C(I
1
,I
k
,I
l
,Q
l
)b
(k)
I
[0]b
(l)
I
[0]b
(k)
Q
[0]b
(l)
Q
[0]


η

K

k=1
C(I
1
,Q
k
)b
(k)
Q
[0] sin θ
(k)

·

K

k
1
=1
K

k
2
=1
K

k

3
=1
a
5
(k
1
,k
2
,k
3
)cosθ
(k
1
)
sin θ
(k
2
)
cos θ
(k
3
)

=
K

k=2

1
2

C(I
1
,Q
k
)
2
d
2
I
b
(k)
Q
[0]
2
−
1
2
C(I
1
,Q
k
)C(Q
1
,I
k
)d
I
d
Q
b

(k)
I
[0]b
(k)
Q
[0]

+

2≤k,l≤K
k=l

1
4
C(I
1
,Q
k
)
2
b
(l)
I
[0]
2
b
(k)
Q
[0]
2

−
1
4
C(I
1
,Q
k
)C(I
1
,I
k
,I
l
,Q
l
)b
(k)
I
[0]b
(l)
I
[0]b
(k)
Q
[0]b
(l)
Q
[0]

η


K

k=1
C(I
1
,Q
k
)b
(k)
Q
[0] sin θ
(k)

·

K

k
1
=1
K

k
2
=1
K

k
3

=1
a
3
(k
1
,k
2
,k
3
)sinθ
(k
1
)
cos θ
(k
2
)
cos θ
(k
3
)

=
K

k=2

1
2
C(I

1
,Q
k
)
2
d
2
I
b
(k)
Q
[0]
2
+
1
2
C(I
1
,Q
k
)
2
d
2
Q
b
(k)
Q
[0]
2

+
NONLINEAR AMPLIFICATION 231
+
1
8
C(I
1
,Q
k
)
2
b
(k)
I
[0]
2
b
(k)
Q
[0]
2
1
8
C(I
1
,Q
k
)
2
b

(k)
Q
[0]
4

+

2≤k,l≤K
k=l

1
4
C(I
1
,Q
k
)
2
b
(l)
I
[0]
2
b
(k)
Q
[0]
2
+
1

4
C(I
1
,Q
k
)
2
b
(k)
Q
[0]
2
b
(l)
Q
[0]
2

η

K

k=1
C(I
1
,Q
k
)b
(k)
Q

[0] sin θ
(k)

·

K

k
1
=1
K

k
2
=1
K

k
3
=1
a
0
(k
1
,k
2
,k
3
)sinθ
(k

1
)
cos θ
(k
2
)
cos θ
(k
3
)

=
K

k=2

3
8
C(I
1
,Q
k
)
2
b
(k)
Q
[0]
4
+

3
8
C(I
1
,Q
k
)
2
b
(k)
I
[0]
2
b
(k)
Q
[0]
2

+

2≤k,l≤K
k=l

3
4
C(I
1
,Q
k

)
2
b
(k)
Q
[0]
2
b
(l)
Q
[0]
2
+
1
4
C(I
1
,Q
k
)
2
b
(l)
I
[0]
2
b
(k)
Q
[0]

2
+
1
2
C(I
1
,Q
k
)C(I
1
,I
k
,I
l
,Q
l
)b
(k)
I
[0]b
(l)
I
[0]b
(k)
Q
[0]b
(l)
Q
[0]


Therefore,
Υ
3
= ηΨ
K

k=1
C(I
1
,I
k
)b
(k)
I
[0] cos θ
(k)
= d
4
I
+ d
2
I
d
2
Q
+
K

k=2


2d
2
I
b
(k)
I
[0]
2
+2d
2
I
b
(k)
Q
[0]
2
+
3
2
C(I
1
,I
k
)
2
d
2
I
b
(k)

I
[0]
2
+ C(I
1
,I
k
)C(Q
1
,Q
k
)d
I
d
Q
b
(k)
I
[0]b
(k)
Q
[0]
+
1
2
C(I
1
,I
k
)

2
d
2
Q
b
(k)
I
[0]
2
+
1
2
C(I
1
,I
k
)
2
b
(k)
I
[0]
4
+
1
2
C(I
1
,I
k

)
2
b
(k)
I
[0]
2
b
(k)
Q
[0]
2

+

2≤k,l≤K
k=l

C(I
1
,I
k
)
2
b
(k)
I
[0]
2
b

(l)
I
[0]
2
+ C(I
1
,I
k
)
2
b
(k)
I
[0]
2
b
(l)
Q
[0]
2

Υ
4
= ηΨ
K

k=1
C(I
1
,Q

k
)b
(k)
Q
[0] sin θ
(k)
=
K

k=2

3
2
C(I
1
,Q
k
)
2
d
2
I
b
(k)
Q
[0]
2
− C(I
1
,Q

k
)C(Q
1
,I
k
)d
I
d
Q
b
(k)
I
[0]b
(k)
Q
[0]
+
1
2
C(I
1
,Q
k
)
2
d
2
Q
b
(k)

Q
[0]
2
+
1
2
C(I
1
,Q
k
)
2
b
(k)
I
[0]
2
b
(k)
Q
[0]
2
+
1
2
C(I
1
,Q
k
)

2
b
(k)
Q
[0]
4

+

2≤k,l≤K
k=l

C(I
1
,Q
k
)
2
b
(l)
I
[0]
2
b
(k)
Q
[0]
2
+ C(I
1

,Q
k
)
2
b
(k)
Q
[0]
2
b
(l)
Q
[0]
2

232 CDMA: ACCESS AND SWITCHING
Following the same method, all the α
2
3
terms and cross-corrleation terms can be
computed similarly.
9A.3 General Results
By computing the first and second moments of S
I
+ D
I
, we can get the following
results:
Y
I

= α
1
d
I
+ α
3
d
3
I
+ α
3
d
I
d
2
Q
+ ηα
1
d
3
Q
+ ηα
1
d
2
I
d
Q
+ ηα
3

d
5
Q
+2ηα
3
d
2
I
d
3
Q
+ ηα
3
d
4
I
d
Q
U
I
=

2α
3
d
I
+2ηα
1
d
Q

+6ηα
3
d
3
Q
+6ηα
3
d
2
I
d
Q


(ζ20k + ζ02k)
+3α
3
d
Q

(ζ40k +2ζ22k + ζ04k)
+12ηα
3
d
Q

2≤k<l≤K
(ζ20k + ζ02k)(ζ20l + ζ02l)
σ
2

I
= α
2
1
(Υ
1
+Υ
2
)+2α
1
α
3
(Υ
3
+Υ
4
)+α
2
3
Υ
5
− S
2
I
+ σ
2
N
σ
2
IQ

=2α
1
α
3
d
I
d
Q

R
2
1k
(ζ20k + ζ02k)
where ζijk, R
ij
,andΥ
1
to Υ
5
are defined as follows.
ζijk =
µ
k
M
M

m=1
cos
i
(2m −1)π

M
sin
j
(2m −1)π
M
(2p)
i+j
S
I
= α
1
d
I
+ α
3
d
3
I
+ α
3
d
I
d
2
Q
+2α
3
d
I


(ζ20k + ζ02k)
Υ
1
= d
2
I
+
1
2

R
2
1k
ζ20k
Υ
2
=
1
2

R
2
1k
ζ02k
Υ
3
= d
4
I
+ d

2
I
d
2
Q
+


2d
2
I
ζ20k +2d
2
I
ζ02k +
3
2
R
2
1k
d
2
I
ζ20k +
1
2
R
2
1k
d

2
Q
ζ20k
+
1
2
R
2
1k
ζ40k +
1
2
R
2
1k
ζ22k

+

2≤k,l≤K
k=l
R
2
1k
ζ20k (ζ20l + ζ02l)
Υ
4
=

R

2
1k

3
2
d
2
I
ζ02k +
1
2
d
2
Q
ζ02k +
1
2
ζ22k +
1
2
ζ04k

+

2≤k,l≤K
k=l
R
2
1k
ζ02l (ζ20k + ζ02k)

Υ
5
=Υ
0 1
+Υ
1 1
+Υ
1 2
+Υ
1 3
+Υ
1 4
+Υ
2 1 1
+Υ
2 1 2
+Υ
2 2 1
+Υ
2 2 2
+Υ
2 3
+Υ
3 1
+Υ
3 2
+Υ
3 3
Υ
0 1

= d
6
I
+2d
4
I
d
2
Q
+ d
2
I
d
4
Q
NONLINEAR AMPLIFICATION 233
Υ
1 1
=


1
2

9d
2
I
+ d
2
Q


(ζ40k + ζ04k)+
1
2

9d
4
I
+ d
4
Q

R
2
1k
(ζ20k+ζ022k)
+
1
2
(ζ60k+ζ06k+ζ24k+ζ42k)R
2
1k
+7d
2
I
ζ22k

Υ
1 2
=



4

d
4
I
+ d
2
I
d
2
Q

(ζ20k + ζ02k)+2d
2
I
ζ22k
+5d
2
I
d
2
Q
R
2
1k
(ζ20k + ζ02k)+3d
2
I

R
2
1k
(ζ40k +2ζ22k + ζ04k)

Υ
1 3
=


−d
2
Q
ζ22k+2d
2
Q
ζ22k

R
2
1k
+1

+d
2
Q
R
2
1k
(ζ40k+ζ04k)


Υ
1 4
=

R
2
1k
(ζ42k + ζ24k)
Υ
2 1 1
=

2≤k,l≤K
k=l

5
2
R
2
1k
(ζ20k + ζ02k)(ζ40l + ζ04l)+d
2
I
ζ20kζ02l

8+10R
2
kl


+6R
2
1k
d
2
I
(ζ20kζ20l + ζ20kζ02l + ζ20lζ02k + ζ02kζ02l)

Υ
2 1 2
=

2≤k<l≤K
d
2
I
(8 + 10R
2
kl
)(ζ20kζ20l + ζ02kζ02l)
Υ
2 2 1
=2

2≤k,l≤K
k=l

d
2
Q

R
2
1k
(ζ20k + ζ02k)(ζ20l + ζ02l)+d
2
Q
R
2
kl
ζ20kζ02l
+ R
2
1k
(ζ40k + ζ04k)(ζ20l + ζ02l)
+2R
2
1k
ζ22k (ζ20l + ζ02l)+2R
2
1l
ζ22kζ02l

Υ
2 2 2
=2d
2
Q

2≤k<l≤K
(ζ02k + ζ20k) ζ02lR

2
kl
Υ
2 3
=3

2≤k,l≤K
k=l
(ζ20k + ζ02k)ζ22lR
2
1k
Υ
3 1
=6

2≤k<l<m≤K
(ζ20kζ20lζ20m + ζ02kζ02lζ02m) R
2
1klm
Υ
3 2
=

2≤l<m≤K
l=k,m=k

6(ζ20kζ20lζ20m + ζ02kζ20lζ20m) R
2
1klm
+4 (ζ20kζ02lζ02m + ζ20kζ20lζ20m

+ζ02kζ20lζ20m + ζ02kζ02lζ02m) R
2
1k

Υ
3 3
=4

2≤k,l,m≤K
l=k,m=k
(ζ20k + ζ02k) ζ20lζ02mR
2
1k
R
ij
=
1
N
N−1

n=0
c
(i)
[n]c
(j)
[n],i= j
R
ijkl
=
1

N
N−1

n=0
c
(i)
[n]c
(j)
[n]c
(k)
[n]c
(l)
[n],i= j = k = l
234 CDMA: ACCESS AND SWITCHING
9A.4 SER Evaluation for MPSK CDMA
Let the outputs of the two branches of the first correlator be
Z
I
= D
I
+ U
I
+ N
I
Z
Q
= D
Q
+ U
Q

+ N
Q
Let Z
I
= ηZ
I
and Z
Q
= ηZ
Q
. The three terms represent the desired signal, other-user
interference plus crosstalk, and AWGN. Then for the joint distribution of (z
I
,z
Q
)we
have
f(z
I
,z
Q
)=
1
2πσ
I
σ
Q

1 −ρ
2

I,Q
exp

−
(z
I
− Z
I
)
2
+(z
Q
− Z
Q
)
2
− 2ρ
I,Q
(z
I
− Z
I
)(z
Q
− Z
Q
)
2(1 −ρ
2
I,Q

)σ
I
σ
Q

The random variables Z
I
and Z
Q
are correlated; we denote their correlation coefficient
by
f(z
I
,z
Q
)dz
I
dz
Q
=
ˆ
f(ρ, φ)ρdρdφ
Joint distribution in polar coordinates can be shown as
ˆ
f(ρ, φ)=
1
2πσ
I
σ
Q


1−ρ
2
I,Q
exp[X
1
]=
1
2πσ
I
σ
Q

1−ρ
2
I,Q
exp[X
2
]
=
1
2πσ
I
σ
Q

1−ρ
2
I,Q
exp






−
[1−ρ
I,Q
sin(2φ)]

ρ−
cos(φ−φ
m
)−ρ
I,Q
sin(φ+φ
m
)
1−ρ
I,Q
sin(2φ)

2
2(1 −ρ
2
I,Q
)σ
I
σ
Q






· exp



−
[1−ρ
I,Q
sin(2φ
m
)]−
[cos(φ−φ
m
)−ρ
I,Q
sin(φ+φ
m
)]
2
1−ρ
I,Q
sin(2φ)
2(1−ρ
2
I,Q
)σ

I
σ
Q



where the exponents X
1
and X
2
are given by
X
1
= −
(ρ cos φ−cos φ
m
)
2
+(ρ sin φ−sin φ
m
)
2
−2ρ
I,Q
(ρ cos φ−cos φ
m
)(ρ sin φ−sin φ
m
)
2(1−ρ

2
I,Q
)σ
I
σ
Q
X
2
= −
[1 − ρ
I,Q
sin(2φ)]ρ
2
−2[cos(φ−φ
m
)−ρ
I,Q
sin(φ+φ
m
)]ρ+[1−ρ
I,Q
sin(2φ
m
)]
2(1−ρ
2
I,Q
)σ
I
σ

Q
Then the probability of error given m is
P
e|m
=1−

φ
m
+
π
M
φ
m
−
π
M


∞
0
ˆ
f(ρ, φ)ρdρ

dφ
NONLINEAR AMPLIFICATION 235
We first compute the integral with respect to ρ in closed form as
1
2πσ
I
σ

Q

1 −ρ
2
I,Q

∞
0
exp





−
[1 −ρ
I,Q
sin(2φ)]

ρ −
cos(φ−φ
m
)−ρ
I,Q
sin(φ+φ
m
)
1−ρ
I,Q
sin(2φ)


2
2

1 −ρ
2
I,Q

σ
I
σ
Q





ρdρ
=
1

2πσ
I
σ
Q

1 −ρ
I,Q
sin(2φ)
cos(φ −φ

m
) −ρ
I,Q
sin(φ + φ
m
)
1 −ρ
I,Q
sin(2φ)
However, the integral with respect to φ cannot be put in closed form. Thus, the final
result is
P
e|m
=1−

φ
m
+
π
M
φ
m
−
π
M
¯
f(φ, φ
m
)dφ
where

¯
f(φ, φ
m
)=
1

2πσ
I
σ
Q

1 −ρ
I,Q
sin(2φ)
cos(φ −φ
m
) −ρ
I,Q
sin(φ + φ
m
)
1 −ρ
I,Q
sin(2φ)
· exp{X}
In the above expression the exponent X is given by
X = −
[1 −ρ
I,Q
sin(2φ

m
)] −
[cos(φ−φ
m
)−ρ
I,Q
sin(φ+φ
m
)]
2
1−ρ
I,Q
sin(2φ)
2(1 −ρ
2
I,Q
)σ
I
σ
Q
where φ
m
=(2m−1)π/M for m =1, 2, ,M. For equi-probable M -ary PSK symbols,
the probability of error is thus
P
e
=
1
M
M


m=1
P
e|m
9A.5 Input Power, Output Power and Input Back-Off
Suppose all users have the same transmitting power, i.e. X
k
∈

±
√
2P

for all k.
A.5.1 Input Power
BPSK:
Y =
K

k=1
X
k
where K is the number of users. The average power at the input of a nonlinear amplifier
is thus
η
1
T
s

T

s
0
(Y cos(ω
c
t))
2
dt =
ηY
2
2
= KP